The $\eta -{\eta }^{\prime }$ mixing mass term due to the derivative coupling $SU(3)\times SU(3)$ symmetry breaking term, produces an additional momentum-dependent pole term for processes with ${\eta }^{\prime }$ but is suppressed in the $\eta$ amplitude by a factor $m_{\eta }^2/m_{{\eta }^{\prime }}^2$. This seems to be the origin of the two-angle description of the pseudoscalar decay constants used in the literature. In this paper, by diagonalizing both the mixing mass term and the momentum-dependent mixing term, we show that the $\eta -{\eta }^{\prime }$ system can be described by a meson field renormalization and a new mixing angle $\theta$ which differs from the usual mixing angle ${\theta }_P$ by a small momentum-dependent mixing $d$ term. This new mixing scheme with exact treatment of the momentum-dependent mixing term is actually simpler than the perturbation treatment and should be used in any determination of the $\eta -{\eta }^{\prime }$ mixing angle and the momentum-dependent mixing term. Assuming nonet symmetry for the ${\eta }_0$ singlet amplitude, from the sum rules relating $\theta$ and $d$ to the measured vector meson radiative decay amplitudes, we obtain consistent solutions: $\theta =-(13.99\pm 3.1)°$, $d=0.12\pm 0.03$ from $\rho \to \eta \gamma$ and ${\eta }^{\prime }\to \rho \gamma$ decays, for $\omega$, $\theta =-(15.47\pm 3.1)°$, $d=0.11\pm 0.03$, and for $\varphi$, $\theta =-(12.66\pm 2.1)°$, $d=0.10\pm 0.03$. It seems that vector meson radiative decays would favor a small $\eta -{\eta }^{\prime }$ mixing angle and a small momentum-dependent mixing term.

• Received 8 July 2015

DOI:https://doi.org/10.1103/PhysRevD.92.054021